1. Chapter 27
Lecture 12:
Circuits

2. Direct Current
When the current in a circuit has a constant direction, the current is called direct current
Most of the circuits analyzed will be assumed to be in steady state, with constant magnitude and direction
Because the potential difference between the terminals of a battery is constant, the battery produces direct current
The battery is known as a source of emf

3. Electromotive Force
The electromotive force (emf), e, of a battery is the maximum possible voltage that the battery can provide between its terminals
The emf supplies energy, it does not apply a force
The battery will normally be the source of energy in the circuit
The positive terminal of the battery is at a higher potential than the negative terminal
We consider the wires to have no resistance

4. Internal Battery Resistance
If the internal resistance is zero, the terminal voltage equals the emf
In a real battery, there is internal resistance, r
The terminal voltage, DV = e – Ir
Use the active figure to vary the emf and resistances and see the effect on the graph

5. EMF, cont
The emf is equivalent to the open-circuit voltage
This is the terminal voltage when no current is in the circuit
This is the voltage labeled on the battery
The actual potential difference between the terminals of the battery depends on the current in the circuit
DV = e – Ir

6. Load Resistance
The terminal voltage also equals the voltage across the external resistance
This external resistor is called the load resistance
In the previous circuit, the load resistance is just the external resistor
In general, the load resistance could be any electrical device
These resistances represent loads on the battery since it supplies the energy to operate the device containing the resistance

7. Power The total power output of the battery is
This power is delivered to the external resistor (I 2 R) and to the internal resistor (I2 r)

8. Resistors in Series
When two or more resistors are connected end-to-end, they are said to be in series
For a series combination of resistors, the currents are the same in all the resistors because the amount of charge that passes through one resistor must also pass through the other resistors in the same time interval
The potential difference will divide among the resistors such that the sum of the potential differences across the resistors is equal to the total potential difference across the combination

9. Resistors in Series, cont
Potentials add
ΔV = IR1 + IR2
= I (R1+R2)
Consequence of Conservation of Energy
The equivalent resistance has the same effect on the circuit as the original combination of resistors

10. Equivalent Resistance – Series
Req = R1 + R2 + R3 + …
The equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any individual resistance
If one device in the series circuit creates an open circuit, all devices are inoperative

11. Some Circuit Notes
A local change in one part of a circuit may result in a global change throughout the circuit
For example, changing one resistor will affect the currents and voltages in all the other resistors and the terminal voltage of the battery
In a series circuit, there is one path for the current to take
In a parallel circuit, there are multiple paths for the current to take

12. Resistors in Parallel
The potential difference across each resistor is the same because each is connected directly across the battery terminals
A junction is a point where the current can split
The current, I, that enters a point must be equal to the total current leaving that point
I = I 1 + I 2
The currents are generally not the same
Consequence of Conservation of Charge

13. Equivalent Resistance – Parallel, Examples
Equivalent resistance replaces the two original resistances

14. Equivalent Resistance – Parallel
The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance
The equivalent is always less than the smallest resistor in the group

15. Resistors in Parallel, Final
In parallel, each device operates independently of the others so that if one is switched off, the others remain on
In parallel, all of the devices operate on the same voltage
The current takes all the paths
The lower resistance will have higher currents
Even very high resistances will have some currents
Household circuits are wired so that electrical devices are connected in parallel

16. Combinations of Resistors
The 8.0-W and 4.0-W resistors are in series and can be replaced with their equivalent, 12.0 W
The 6.0-W and 3.0-W resistors are in parallel and can be replaced with their equivalent, 2.0 W
These equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0 W

17. Kirchhoff’s Rules
There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor
Two rules, called Kirchhoff’s rules, can be used instead

18. Kirchhoff’s Junction Rule
The sum of the currents at any junction must equal zero
Currents directed into the junction are entered into the -equation as +I and those leaving as -I
A statement of Conservation of Charge
Mathematically,

19. More about the Junction Rule
I1 - I2 - I3 = 0
Required by Conservation of Charge
Diagram (b) shows a mechanical analog

20. Kirchhoff’s Loop Rule Loop Rule Mathematically,
The sum of the potential differences across all elements around any closed circuit loop must be zero
A statement of Conservation of Energy
Mathematically,

21. More about the Loop Rule
Traveling around the loop from a to b
In (a), the resistor is traversed in the direction of the current, the potential across the resistor is – IR
In (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is is + IR

22. Loop Rule, final
In (c), the source of emf is traversed in the direction of the emf (from – to +), and the change in the electric potential is +ε
In (d), the source of emf is traversed in the direction opposite of the emf (from + to -), and the change in the electric potential is -ε

23. Junction Equations from Kirchhoff’s Rules
Use the junction rule as often as needed, so long as each time you write an equation, you include in it a current that has not been used in a previous junction rule equation
In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

24. Loop Equations from Kirchhoff’s Rules
The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation
You need as many independent equations as you have unknowns

25. Kirchhoff’s Rules Equations, final
In order to solve a particular circuit problem, the number of independent equations you need to obtain from the two rules equals the number of unknown currents
Any capacitor acts as an open branch in a circuit
The current in the branch containing the capacitor is zero under steady-state conditions

26. A multi-loop Circuit

27. RC Circuits
In direct current circuit containing capacitors, the current may vary with time
The current is still in the same direction
An RC circuit will contain a series combination of a resistor and a capacitor

28. RC Circuit Adjust the values of R and C
Observe the effect on the charging and discharging of the capacitor

29. Charging a Capacitor
When the circuit is completed, the capacitor starts to charge
The capacitor continues to charge until it reaches its maximum charge (Q = Cε)
Once the capacitor is fully charged, the current in the circuit is zero

30. Charging an RC Circuit, cont.
As the plates are being charged, the potential difference across the capacitor increases
At the instant the switch is closed, the charge on the capacitor is zero
Once the maximum charge is reached, the current in the circuit is zero
The potential difference across the capacitor matches that supplied by the battery

31. Charging a Capacitor in an RC Circuit
The charge on the capacitor varies with time
q(t) = Ce(1 – e-t/RC)
= Q(1 – e-t/RC)
The current can be found
t is the time constant
 = RC

32. Time Constant, Charging
The time constant represents the time required for the charge to increase from zero to 63.2% of its maximum
t has units of time
The energy stored in the charged capacitor is ½ Qe = ½ Ce2

33. Discharging a Capacitor in an RC Circuit
When a charged capacitor is placed in the circuit, it can be discharged
q(t) = Qe-t/RC
The charge decreases exponentially

34. Discharging Capacitor
At t =  = RC, the charge decreases to Qmax
In other words, in one time constant, the capacitor loses 63.2% of its initial charge
The current can be found
Both charge and current decay exponentially at a rate characterized by t = RC

35. Example, Discharging an RC circuit :

36. Galvanometer
A galvanometer is the main component in analog meters for measuring current and voltage
Digital meters are in common use
Digital meters operate under different principles

37. Galvanometer, cont
A galvanometer consists of a coil of wire mounted so that it is free to rotate on a pivot in a magnetic field
The field is provided by permanent magnets
A torque acts on a current in the presence of a magnetic field

38. Galvanometer, final The torque is proportional to the current
The larger the current, the greater the torque
The greater the torque, the larger the rotation of the coil before the spring resists enough to stop the rotation
The deflection of a needle attached to the coil is proportional to the current
Once calibrated, it can be used to measure currents or voltages

39. Ammeter An ammeter is a device that measures current
The ammeter must be connected in series with the elements being measured
The current must pass directly through the ammeter

40. Ammeter in a Circuit
The ammeter is connected in series with the elements in which the current is to be measured
Ideally, the ammeter should have zero resistance so the current being measured is not altered

41. Ammeter from Galvanometer
The galvanometer typically has a resistance of 60 W
To minimize the resistance, a shunt resistance, Rp, is placed in parallel with the galvanometer

42. Constructing An Ammeter
Predict the value of the shunt resistor, Rs, needed to achieve full scale deflection on the meter
Use the active figure to test your result

43. Ammeter, final
The value of the shunt resistor must be much less than the resistance of the galvanometer
Remember, the equivalent resistance of resistors in parallel will be less than the smallest resistance
Most of the current will go through the shunt resistance, this is necessary since the full scale deflection of the galvanometer is on the order of 1 mA

44. Voltmeter A voltmeter is a device that measures potential difference
The voltmeter must be connected in parallel with the elements being measured
The voltage is the same in parallel

45. Voltmeter in a Circuit
The voltmeter is connected in parallel with the element in which the potential difference is to be measured
Polarity must be observed
Ideally, the voltmeter should have infinite resistance so that no current would pass through it
Corrections can be made to account for the known, non-infinite, resistance of the voltmeter

46. Voltmeter from Galvanometer
The galvanometer typically has a resistance of 60 W
To maximize the resistance, another resistor, Rs, is placed in series with the galvanometer
Calculate Rs and use the active figure to check your result

47. Voltmeter, final
The value of the added resistor must be much greater than the resistance of the galvanometer
Remember, the equivalent resistance of resistors in series will be greater than the largest resistance
Most of the current will go through the element being measured, and the galvanometer will not alter the voltage being measured

48. Household Wiring
The utility company distributes electric power to individual homes by a pair of wires
Each house is connected in parallel with these wires
One wire is the “live wire” and the other wire is the neutral wire connected to ground

49. Household Wiring, final
A meter is connected in series with the live wire entering the house
This records the household’s consumption of electricity
After the meter, the wire splits so that multiple parallel circuits can be distributed throughout the house
Each circuit has its own circuit breaker
For those applications requiring 240 V, there is a third wire maintained at 120 V below the neutral wire

50. Short Circuit
A short circuit occurs when almost zero resistance exists between two points at different potentials
This results in a very large current
In a household circuit, a circuit breaker will open the circuit in the case of an accidental short circuit
This prevents any damage
A person in contact with ground can be electrocuted by touching the live wire